Residual ISI Obtained by Blind Adaptive Equalizers and Fractional Noise

Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 972174, 11 pages http://dx.doi.org/10.1155/2013/972174

Research Article Residual ISI Obtained by Blind Adaptive Equalizers and Fractional Noise Monika Pinchas Department of Electrical and Electronic Engineering, Ariel University of Samaria, Ariel 40700, Israel Correspondence should be addressed to Monika Pinchas; [email protected] Received 27 March 2013; Accepted 28 April 2013 Academic Editor: Ming Li Copyright © 2013 Monika Pinchas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Recently, a closed-form approximated expression was derived by the same author for the achievable residual intersymbol interference (ISI) case that depends on the step-size parameter, equalizer’s tap length, input signal statistics, signal-to-noise ratio (SNR), and channel power. But this expression was obtained by assuming that the input noise is a white Gaussian process where the Hurst exponent (H) is equal to 0.5. In this paper, we derive a closed-form approximated expression (or an upper limit) for the residual ISI obtained by blind adaptive equalizers valid for fractional Gaussian noise (fGn) input where the Hurst exponent is in the region of 0.5 ≤ 𝐻 < 1. Up to now, the statistical behaviour of the residual ISI was not investigated. Furthermore, the convolutional noise for the latter stages of the deconvolutional process was assumed to be a white Gaussian process (𝐻 = 0.5). In this paper, we show that the Hurst exponent of the residual ISI is close to one, almost independent of the SNR or equalizer’s tap length but depends on the step-size parameter. In addition, the convolutional noise obtained in the steady state is a noise process having a Hurst exponent depending on the step-size parameter.

1. Introduction We consider a blind deconvolution problem in which we observe the output of an unknown, possibly nonminimum phase, linear system (SISO-FIR system) from which we want to recover its input (source) using an adjustable linear filter (equalizer). The problem of blind deconvolution arises comprehensively in various applications such as digital communications, seismic signal processing, speech modeling and synthesis, ultrasonic nondestructive evaluation, and image restoration [1]. Blind deconvolution algorithms are essentially adaptive filtering algorithms designed such that they do not require the external supply (training sequence) of a desired response to generate the error signal in the output of the adaptive equalization filter [2, 3]. The algorithm itself generates an estimate of the desired response by applying a nonlinear transformation to sequences involved in the adaptation process [2, 3]. Let us consider for a moment the digital communication case. During transmission,

a source signal undergoes a convolutive distortion between its symbols and the channel impulse response. This distortion is referred to as ISI. Thus, a blind adaptive equalizer is used to remove the convolutive effect of the system to produce the source signal [2, 4–6]. Up to recently [7, 8], the performance of a chosen equalizer (the achievable residual ISI) could be obtained only via simulation. The equalization performance depends on the nature of the chosen equalizer (on the memoryless nonlinearity situated at the output of the equalizer’s filter), on the channel characteristics, on the added noise, on the step-size parameter used in the adaptation process, on the equalizer’s tap length, and on the input signal statistics. Fast convergence speed and reaching a residual ISI where the eye diagram is considered to be open (for the communication case) are the main requirements from a blind equalizer. Fast convergence speed may be obtained by increasing the stepsize parameter. But increasing the step-size parameter may lead to a higher residual ISI which might not meet any

2

Mathematical Problems in Engineering 𝑤(𝑛) Adaptive equalizer + 𝑥(𝑛)

ℎ(𝑛)

+

𝑦(𝑛)

𝑐(𝑛)

𝑧(𝑛)

(1) The input sequence 𝑥(𝑛) belongs to a two independent quadrature carrier case constellation input with variance 𝜎𝑥2 , where 𝑥𝑟 (𝑛) and 𝑥𝑖 (𝑛) are the real and imaginary parts of 𝑥(𝑛), respectively, and 𝜎𝑥2𝑟 is the variance of 𝑥𝑟 (𝑛). In the following we denote 𝑥𝑟 (𝑛) as 𝑥𝑟 .

Figure 1: Block diagram of a baseband communication system.

more the system’s requirements. Up to recently, the system designer had to carry out, for a given channel and type of equalizer, many simulations in order to obtain the optimal step-size parameter for a required residual ISI. Recently [7, 8], a closed-form approximated expression was derived by the same author for the achievable residual ISI case that depends on the step-size parameter, equalizer’s tap length, input signal statistics, signal-to-noise ratio (SNR), and channel power. But this expression was obtained by assuming that the input noise is a white Gaussian process where the Hurst exponent (𝐻) is equal to 0.5. A white Gaussian process is a special case (𝐻 = 0.5) of the fractional Gaussian noise (fGn) model [9]. FGn with 𝐻 ∈ (0.5, 1) corresponds to the case of long-range dependency (LRD) [9]. LRD implies heavy-tailed probability density functions, which in general imply more random, see [10–13]. This point of view was recently detailed by [14, 15]. An fGn is a generalization of ordinary white Gaussian noise, and it is a versatile model for broadband noise dominated by no particular frequency band [16]. For 0.5 < 𝐻 < 1, fGn is bandlimited to (−𝜔𝑐 , 𝜔𝑐 ) at level 𝜀1 > 0 [17]. In this paper, we derive a closed-form approximated expression (or an upper limit) for the residual ISI obtained by blind adaptive equalizers valid for fGn input where the Hurst exponent is in the region of 0.5 ≤ 𝐻 < 1. Please note, 𝐻 = 1 is the limit case, which does not have much practical sense [18, 19]. Up to now, the statistical behaviour of the residual ISI was not investigated. Furthermore, the convolutional noise for the latter stages of the deconvolutional process was assumed to be a white Gaussian process (𝐻 = 0.5). In this paper, we show that the Hurst exponent of the residual ISI is close to one, almost independent of the SNR or equalizer’s tap length but depends on the step-size parameter. In addition, the convolutional noise obtained in the steady state is a noise process with a Hurst exponent depending on the step-size parameter. The paper is organized as follows. After having described the system under consideration in Section 2, the closed-form approximated expression (or upper limit) for the residual ISI is introduced in Section 3. In Section 4 the statistical behavior of the residual ISI and convolutional noise are presented and the conclusion is given in Section 5.

(2) The unknown channel ℎ(𝑛) is a possibly nonminimum phase linear time-invariant filter in which the transfer function has no “deep zeros”; namely, the zeros lie sufficiently far from the unit circle. (3) The equalizer 𝑐(𝑛) is a tap-delay line. (4) The noise 𝑤(𝑛) consists of 𝑤(𝑛) = 𝑤𝑟 (𝑛) + 𝑗𝑤𝑖 (𝑛), where 𝑤𝑟 (𝑛) and 𝑤𝑖 (𝑛) are the real and imaginary parts of 𝑤(𝑛), respectively, and 𝑤𝑟 (𝑛) and 𝑤𝑖 (𝑛) are independent. Both 𝑤𝑟 (𝑛) and 𝑤𝑖 (𝑛) are fractional Gaussian noises (fGn) with zero mean. Consider 𝜎𝑤2 𝑟 = 𝐸[𝑤𝑟2 (𝑛)], 𝜎𝑤2 𝑖 = 𝐸[𝑤𝑖2 (𝑛)], 𝐸[𝑤𝑟 (𝑛 − 𝑘)𝑤𝑟 (𝑛 − 𝑚)] = (𝜎𝑤2 𝑟 /2)[(|𝑚 − 𝑘| − 1)2𝐻 − 2(|𝑚 − 𝑘|)2𝐻 +(|𝑚 − 𝑘| + 1)2𝐻], and 𝐸[𝑤𝑖 (𝑛−𝑘)𝑤𝑖 (𝑛− 𝑚)] = (𝜎𝑤2 𝑖 /2)[(|𝑚 − 𝑘| − 1)2𝐻 − 2(|𝑚 − 𝑘|)2𝐻 + (|𝑚 − 𝑘| + 1)2𝐻], where 𝐸[⋅] denotes the expectation operator on (⋅) and 𝐻 is the Hurst exponent. The transmitted sequence 𝑥(𝑛) is sent through the channel ℎ(𝑛) and is corrupted with noise 𝑤(𝑛). Therefore, the equalizer’s input sequence 𝑦(𝑛) may be written as

𝑦 (𝑛) = 𝑥 (𝑛) ∗ ℎ (𝑛) + 𝑤 (𝑛) ,

where “∗” denotes the convolution operation. The equalized output signal can be written as:

̃ (𝑛) , 𝑧 (𝑛) = 𝑥 (𝑛) + 𝑝 (𝑛) + 𝑤

(2)

where 𝑝(𝑛) is the convolutional noise, namely, the residual intersymbol interference (ISI) arising from the difference between the ideal equalizer’s coefficients and those chosen in ̃(𝑛) = 𝑤(𝑛) ∗ 𝑐(𝑛). The ISI is often used as the system and 𝑤 a measure of performance in equalizers’ applications, defined by

2. System Description The system under consideration is illustrated in Figure 1, where we make the following assumptions.

(1)

ISI =

󵄨 󵄨2 ∑𝑚̃ 󵄨󵄨󵄨𝑠̃𝑚̃ 󵄨󵄨󵄨 − |̃𝑠|2max |̃𝑠|2max

,

(3)

Mathematical Problems in Engineering

3

where |̃𝑠|max is the component of 𝑠̃, given in (4), having the maximal absolute value

or 𝑚𝑝

𝑚𝑝

𝑚𝑝1

𝑚𝑝 = max [Sol1 1 , Sol2 1 ] 𝑠̃ (𝑛) = 𝑐 (𝑛) ∗ ℎ (𝑛) = 𝛿 (𝑛) + 𝜁 (𝑛) ,

(4)

< 0, (8)

𝑚𝑝1

=

𝑚𝑝1

=

Sol2

−𝐵1 + √𝐵12 − 4𝐴 1 𝐶1 𝐵 2𝐴 1 −𝐵1 − √𝐵12 − 4𝐴 1 𝐶1 𝐵 2𝐴 1

,

,

𝐴1 2 = (𝐵 (45𝜎𝑥2𝑟 𝑎32 + 18𝜎𝑥2𝑟 𝑎3 𝑎12 + 6𝑎1 𝑎3 + 9𝜎𝑥2𝑟 𝑎12 + 2𝑎1 𝑎12 ) 2 −2 (3𝑎3 + 𝑎12 )) + 𝐵 (45𝑎32 + 18𝑎3 𝑎12 + 9𝑎12 ) 𝜎𝑤2̃𝑟 , 2

2

2 + 12𝜎𝑥2𝑟 𝑎1 𝑎3 𝐵1 = (𝐵 (12(𝜎𝑥2𝑟 ) 𝑎3 𝑎12 + 6(𝜎𝑥2𝑟 ) 𝑎12

+ 4𝜎𝑥2𝑟 𝑎1 𝑎12 + 𝑎12 + 15𝐸 [𝑥𝑟4 ] 𝑎32 2 +2𝐸 [𝑥𝑟4 ] 𝑎3 𝑎12 + 𝐸 [𝑥𝑟4 ] 𝑎12 )

−2 (𝑎1 + 3𝜎𝑥2𝑟 𝑎3 + 𝜎𝑥2𝑟 𝑎12 ) )

𝑐 eq (𝑛 + 1) = 𝑐 eq (𝑛) + 𝜇 ⋅ (−∇𝑐eq 𝐹 (𝑛)) = 𝑐 eq (𝑛) − 𝜇

𝑚𝑝1

⋅ Sol2

where

Sol1 where 𝛿 is the Kronecker delta function and 𝜁(𝑛) stands for the difference (error) between the ideal and the actual value used for 𝑐(𝑛). Although the ISI is often used as a measure of performance in equalizers’ applications, its statistical properties were never investigated. Next we turn to the adaptation mechanism of the equalizer which is based on a predefined cost function 𝐹(𝑛) that characterizes the intersymbol interference, see [20–24]. Minimizing this 𝐹(𝑛) with respect to the equalizer parameters will reduce the convolutional error. Minimization is performed with the gradient descent algorithm that searches for an optimal filter tap setting by moving in the direction of the negative gradient −∇𝑐 𝐹(𝑛) over the surface of the cost function in the equalizer filter tap space [25]. Thus the updated equation is given by [25]

for Sol1

(5)

𝜕𝐹 (𝑛) ∗ 𝑦 (𝑛) , 𝜕𝑧 (𝑛)

2 + 𝐵 (45𝑎32 + 16𝑎3 𝑎12 + 9𝑎12 ) 𝜎𝑤4̃𝑟

+ (𝐵 (90𝑎32 𝜎𝑥2𝑟 + 36𝑎3 𝑎12 𝜎𝑥2𝑟 + 12𝑎1 𝑎3 2 2 +18𝑎12 𝜎𝑥𝑟 + 4𝑎1 𝑎12 ) − 2𝑎12 − 6𝑎3 ) 𝜎𝑤2̃𝑟 ,

where 𝜇 is the step-size parameter and 𝑐 eq (𝑛) is the equalizer vector where the input vector is 𝑦(𝑛) = [𝑦(𝑛) ⋅ ⋅ ⋅ 𝑦(𝑛−𝑁+1)]𝑇 and 𝑁 is the equalizer’s tap length. The operator (⋅)𝑇 denotes for transpose of the function (⋅). Recently [8], a closed-form approximated expression was derived for the achievable residual ISI case (based on [7]) that depends on the step-size parameter, equalizer’s tap length, input signal statistics, SNR, and channel power and is given by

2

𝐶1 = (2(𝜎𝑥2𝑟 ) 𝑎1 𝑎12 + 𝜎𝑥2𝑟 𝑎12 + 2𝐸 [𝑥𝑟4 ] 𝜎𝑥2𝑟 𝑎3 𝑎12 2 +𝐸 [𝑥𝑟4 ] 𝜎𝑥2𝑟 𝑎12 + 2𝐸 [𝑥𝑟4 ] 𝑎1 𝑎3 + 𝐸 [𝑥𝑟6 ] 𝑎32 ) 2 + (15𝑎32 + 6𝑎3 𝑎12 + 3𝑎12 ) 𝜎𝑤6̃𝑟

+ (45𝑎32 𝜎𝑥2𝑟 + 18𝑎3 𝑎12 𝜎𝑥2𝑟 + 6𝑎1 𝑎3 2 2 +9𝑎12 𝜎𝑥𝑟 + 2𝑎1 𝑎12 ) 𝜎𝑤4̃𝑟

+ (𝑎12 + 12𝑎1 𝑎3 𝜎𝑥2𝑟 + 4𝑎1 𝑎12 𝜎𝑥2𝑟 + 15𝑎32 𝐸 [𝑥𝑟4 ] ISI = 10 log10 (𝑚𝑝 ) − 10 log10 (𝜎𝑥2𝑟 ) ,

(6)

2

2 𝐸 [𝑥𝑟4 ] + 12𝑎3 𝑎12 (𝜎𝑥2𝑟 ) + 2𝑎3 𝑎12 𝐸 [𝑥𝑟4 ] + 𝑎12 2

2 (𝜎𝑥2𝑟 ) ) 𝜎𝑤2̃𝑟 , +6𝑎12

where 𝜎𝑥2𝑟 is the variance of the real part of the input sequence 𝑥(𝑛) and 𝑚𝑝 is defined by

2 𝑘𝑘=𝑅−1 󵄨2 𝜇𝑁𝜎𝑥 󵄨 𝐵 = 𝜇𝑁𝜎𝑥2 ∑ 󵄨󵄨󵄨ℎ𝑘𝑘 (𝑛)󵄨󵄨󵄨 + , SNR 𝑘𝑘=0

(9) 𝑚𝑝

𝑚𝑝

𝑚𝑝 = min [Sol1 1 , Sol2 1 ]

𝑚𝑝1

for Sol1

𝑚𝑝1

> 0, Sol2

>0 (7)

𝜎𝑤2̃𝑟

=

𝜎𝑥2𝑟

󵄨2 󵄨󵄨 SNR ∑𝑘𝑘=𝑅−1 󵄨󵄨ℎ𝑘𝑘 (𝑛)󵄨󵄨󵄨 𝑘𝑘=0

.

(10)

4


The channel length is 𝑅, 𝜎𝑤2̃𝑟 is the variance of the real part ̃(𝑛), SNR = 𝜎𝑥2 /𝜎𝑤2 , and 𝑎1 , 𝑎12 , 𝑎3 are properties of the of 𝑤 chosen equalizer and are found by Re (

𝜕𝐹 (𝑛) 3 2 ) = (𝑎1 (𝑧𝑟 ) + 𝑎3 (𝑧𝑟 ) + 𝑎12 (𝑧𝑟 ) (𝑧𝑖 ) ) , 𝜕𝑧 (𝑛)

(11)

where Re(⋅) is the real part of (⋅) and 𝑧𝑟 , 𝑧𝑖 are the real and imaginary parts of the equalized output 𝑧(𝑛), respectively. It should be pointed out that the closed-form approximated expression for the residual ISI [8] was obtained by assuming that the noise 𝑤(𝑛) is an additive Gaussian white noise with zero mean and variance 𝜎𝑤2 = 𝐸[𝑤(𝑛)𝑤∗ (𝑛)]. Therefore, it is not applicable for the fGn case (for 𝐻 > 0.5). Thus, a new expression for the achievable residual ISI is needed.

Comments. Please note that for 𝐻 = 0.5 (Gaussian white noise case), the expression for 𝜎𝑤2̃𝑟 given in (12) and (10) is equivalent. By repeating the steps in [8] for the calculation of the expression of the residual ISI, the only place where the difference between the assumption of 𝑤𝑟 (𝑛) and 𝑤𝑖 (𝑛) being Gaussian white noises or fractional Gaussian noises has a major role on the total result of the approximated derived expression for the residual ISI, is in the calculation of 𝜎𝑤2̃𝑟 . Thus, we bring here only the various steps that led to (12). It should be pointed out that assumptions (1)–(6) from above are precisely the same assumptions made in [8]. ̃(𝑛), namely, 𝑤 ̃𝑟 (𝑛), may be expressed Proof. The real part of 𝑤 as 𝑘=𝑁−1

̃𝑟 (𝑛) = ∑ 𝑐𝑘 (𝑛) 𝑤𝑟 (𝑛 − 𝑘) . 𝑤

3. Residual ISI for Fractional Gaussian Noise Input In this section, a closed-form approximated expression (or an upper limit) is derived for the residual ISI valid for the fGn input case supported by simulation results. 3.1. Derivation of the Residual ISI

̃𝑟 (𝑛) is given by Thus, the variance of 𝑤 𝑘=𝑁−1

𝑚=𝑁−1

𝑘=0

𝑚=0

𝜎𝑤2̃𝑟 = 𝐸 [ ∑ 𝑐𝑘 (𝑛) 𝑤𝑟 (𝑛 − 𝑘) ∑ 𝑐𝑚 (𝑛) 𝑤𝑟 (𝑛 − 𝑚)] 𝑘=𝑁−1 𝑚=𝑁−1

= ∑

Theorem 1. Consider the following assumptions.

𝑘=0

(1) The convolutional noise 𝑝(𝑛) is a zero mean, white Gaussian process with variance 𝜎𝑝2 = 𝐸[𝑝(𝑛)𝑝∗ (𝑛)]. The real part of 𝑝(𝑛) is denoted as 𝑝𝑟 (𝑛) and 𝐸[𝑝𝑟2 (𝑛)] = 𝑚𝑝 . (2) The source signal 𝑥(𝑛) is a rectangular QAM (Quadrature Amplitude Modulation) signal (where the real part of 𝑥(𝑛) is independent of the imaginary part of 𝑥(𝑛)) with known variance and higher moments. (3) The convolutional noise 𝑝(𝑛) and the source signal are independent. (4) 𝜕𝐹(𝑛)/𝜕𝑧(𝑛) can be expressed as a polynomial function of the equalized output, namely, as 𝑃(𝑧) of order three. (5) The gain between the source and equalized output signal is equal to one.

which can be also written as 𝑘=𝑁−1

𝜎𝑤2̃𝑟 = 𝜎𝑤2 𝑟 ∑ 𝑐𝑘2 (𝑛) 𝑘=0

𝑘=𝑁−1

+

(15)

𝑐𝑘 (𝑛) 𝑐𝑚 (𝑛) 𝐸

× [𝑤𝑟 (𝑛 − 𝑘) 𝑤𝑟 (𝑛 − 𝑚)] . Next, by using the assumption that the gain between the source and equalized output signal is equal to one, we may write the following expression: 𝑘=𝑁−1

∑ 𝑐𝑘2 (𝑛) ≃

𝑘=0

(9) The Hurst exponent is in the range of 0.5 ≤ 𝐻 < 1.

1 ∑𝑘𝑘=𝑅−1 𝑘𝑘=0

2 ℎ𝑘𝑘 (𝑛)

.

(16)

By substituting (16) into (15) we obtain

The residual ISI expressed in dB units may be defined as (6), (8), and (9) where 𝑅 is the channel length, SNR = 𝜎𝑥2 /𝜎𝑤2 , and 𝑎1 , 𝑎12 , 𝑎3 are properties of the chosen equalizer and one found by (11) and

2 SNR ∑𝑘𝑘=𝑅−1 ℎ𝑘𝑘 [𝑛] 𝑘𝑘=0

∑

𝑘=0,𝑘 ≠ 𝑚 𝑚=0,𝑘 ≠ 𝑚

(8) The channel ℎ(𝑛) has real coefficients.

≅

𝑚=𝑁−1

∑

(7) The added noise is fGn with zero mean.

𝜎𝑤2̃𝑟

∑ 𝑐𝑘 (𝑛) 𝑐𝑚 (𝑛) 𝐸 [𝑤𝑟 (𝑛 − 𝑘) 𝑤𝑟 (𝑛 − 𝑚)]

𝑚=0

(14)

̃(𝑛). (6) The convolutional noise 𝑝(𝑛) is independent of 𝑤

𝜎𝑥2𝑟

(13)

𝑘=0

[1 + √(𝑁 − 1)𝐻 (2𝐻 − 1)] . (12)

𝜎𝑤2̃𝑟 = 𝜎𝑤2 𝑟 +

1 ∑𝑘𝑘=𝑅−1 𝑘𝑘=0


𝑘=𝑁−1

𝑚=𝑁−1

∑

∑

𝑘=0,𝑘 ≠ 𝑚 𝑚=0,𝑘 ≠ 𝑚

𝑐𝑘 (𝑛) 𝑐𝑚 (𝑛) 𝐸

× [𝑤𝑟 (𝑛 − 𝑘) 𝑤𝑟 (𝑛 − 𝑚)] .

(17)


5

The expression (17) can be upper limited by 𝜎𝑤2̃𝑟 ≤ 𝜎𝑤2 𝑟

Now, since we deal with the rectangular QAM case and where ̃𝑖 , we obtain that 𝜎𝑥2𝑟 = 𝜎𝑥2𝑖 and 𝜎𝑤2 𝑟 = ̃𝑟 is independent of 𝑤 𝑤 2 2 𝜎𝑤𝑖 where 𝜎𝑥𝑖 is the variance of the imaginary part of 𝑥(𝑛). Therefore, we may write

1 2 ∑𝑘𝑘=𝑅−1 ℎ𝑘𝑘 (𝑛) 𝑘𝑘=0

+ max [𝐸 [𝑤𝑟 (𝑛 − 𝑘) 𝑤𝑟 (𝑛 − 𝑚)]]

SNR =

󵄨󵄨 𝑘=𝑁−1 󵄨󵄨 𝑐𝑘 (𝑛) 𝑐𝑚 (𝑛)󵄨󵄨󵄨󵄨 ≤ (𝑁 − 1) ∑ 𝑐𝑘2 (𝑛) . 󵄨󵄨 𝑚=0,𝑘 ≠ 𝑚 𝑘=0 󵄨

𝜎𝑤2̃𝑟

𝑚=𝑁−1

∑

󵄨󵄨 󵄨󵄨 𝑁−1 . ∑ 𝑐𝑘 (𝑛) 𝑐𝑚 (𝑛)󵄨󵄨󵄨󵄨 ≤ 𝑘𝑘=𝑅−1 2 󵄨 (𝑛) 󵄨󵄨 ∑𝑘𝑘=0 ℎ𝑘𝑘 𝑚=0,𝑘 ≠ 𝑚 𝑚=𝑁−1

(20)

Thus having 𝜎𝑤2̃𝑟

≤

𝜎𝑤2 𝑟

1 2 ∑𝑘𝑘=𝑅−1 ℎ𝑘𝑘 (𝑛) 𝑘𝑘=0

𝑁−1 ∑𝑘𝑘=𝑅−1 𝑘𝑘=0


𝜎𝑤2 𝑟

, where (21)

[(|𝑚 − 𝑘| − 1)2𝐻 − 2(|𝑚 − 𝑘|)2𝐻

2

2 SNR ∑𝑘𝑘=𝑅−1 ℎ𝑘𝑘 (𝑛) 𝑘𝑘=0

+(|𝑚 − 𝑘| + 1)2𝐻] .

𝜎𝑥2𝑟 2 SNR ∑𝑘𝑘=𝑅−1 ℎ𝑘𝑘 (𝑛) 𝑘𝑘=0

≤

𝜎𝑤2 𝑟

𝑐𝑚 (𝑛 + 1) = 𝑐𝑚 (𝑛) − 𝜇𝐺 (|𝑧 (𝑛)|2 −

2 ∑𝑘𝑘=𝑅−1 ℎ𝑘𝑘 (𝑛) 𝑘𝑘=0

𝑁−1 [(|𝑚 − 𝑘| − 1)2𝐻 max 2 𝑘 ≠ 𝑚,𝑘,𝑚=0:𝑁−1 2𝐻

−2(|𝑚 − 𝑘|)

𝐸 [|𝑥 (𝑛)|2 ]

)

(28)

where 𝜇𝐺 is the step-size. The values for 𝑎1 , 𝑎12 , and 𝑎3 corresponding to Godard’s [20] algorithm are given by (22) 𝑎1 = −

2𝐻

+ (|𝑚 − 𝑘| + 1) ] ] .

𝐸 [|𝑥 (𝑛)|4 ] 𝐸 [|𝑥 (𝑛)|2 ]

,

𝑎12 = 1,

𝑎3 = 1.

(29)

The following two channels were considered.

According to [17], 0.5 [(|𝑚 − 𝑘| − 1)2𝐻 − 2(|𝑚 − 𝑘|)2𝐻 + (|𝑚 − 𝑘| + 1)2𝐻] ≃ 𝐻 (2𝐻 − 1) |𝑚 − 𝑘|2𝐻−2 .

(23)

By substituting (23) into (22) and taking into account that max𝑘 ≠ 𝑚,𝑘,𝑚=0:𝑁−1 |𝑚 − 𝑘|2𝐻−2 = 1 for 0.5 ≤ 𝐻 < 1, we obtain 𝜎𝑤2̃𝑟 ≤ 𝜎𝑤2 𝑟

𝐸 [|𝑥 (𝑛)|4 ]

× 𝑧 (𝑛) 𝑦∗ (𝑛 − 𝑚) ,

1

× [1 +

[1 + √𝑁 − 1𝐻 (2𝐻 − 1)] . (27)

3.2. Simulation. In this section we test our new proposed expression for the residual ISI for the 16QAM case (a modulation using ±{1, 3} levels for in-phase and quadrature components) with Godard’s algorithm [20] for two different SNR and equalizer’s tap length values and for two different channel types. The equalizer taps for Godard’s algorithm [20] were updated according to

This expression (21) can be therefore written as 𝜎𝑤2̃𝑟

[1 + (𝑁 − 1) 𝐻 (2𝐻 − 1)] . (26)

This completes our proof.

𝐸 [𝑤𝑟 (𝑛 − 𝑘) 𝑤𝑟 (𝑛 − 𝑚)] =

𝜎𝑥2𝑟

It turned out, according to simulation results, that (26) is too far away from the averaged residual ISI. Thus, it can not serve in practice as an upper limit for the expected residual ISI. In reality, we may find approximately only around √𝑁 − 1 coefficients in the equalizer’s tap length that may have significant values. Therefore, we may write 𝜎𝑤2̃𝑟 ≃

+ max [𝐸 [𝑤𝑟 (𝑛 − 𝑘) 𝑤𝑟 (𝑛 − 𝑚)]] ×

≤

(19)

Now, by substituting (16) into (19) we obtain 󵄨󵄨 𝑘=𝑁−1 󵄨󵄨 󵄨󵄨 ∑ 󵄨󵄨 󵄨󵄨𝑘=0,𝑘 ≠ 𝑚 󵄨

(25)

By substituting (25) into (24) we obtain:

With the help of the Holder Inequality [26], we may write 󵄨󵄨 𝑘=𝑁−1 󵄨󵄨 󵄨󵄨 ∑ 󵄨󵄨 󵄨󵄨𝑘=0,𝑘 ≠ 𝑚 󵄨

2𝜎𝑥2𝑟 𝜎𝑥2𝑟 𝜎𝑥2 = = . 𝜎𝑤2 2𝜎𝑤2 𝑟 𝜎𝑤2 𝑟

(18)

󵄨󵄨 󵄨󵄨 𝑘=𝑁−1 𝑚=𝑁−1 󵄨󵄨 󵄨󵄨 󵄨 × 󵄨󵄨󵄨 ∑ ∑ 𝑐𝑘 (𝑛) 𝑐𝑚 (𝑛)󵄨󵄨󵄨󵄨 . 󵄨󵄨 󵄨󵄨𝑘=0,𝑘 ≠ 𝑚 𝑚=0,𝑘 ≠ 𝑚 󵄨 󵄨

1 ∑𝑘𝑘=𝑅−1 𝑘𝑘=0


[1 + (𝑁 − 1) 𝐻 (2𝐻 − 1)] . (24)

Channel 1 (initial 𝐼𝑆𝐼 = 0.44). The channel parameters were determined according to [24]: ℎ𝑛 = (0 for 𝑛 < 0; −0.4 for 𝑛 = 0 0.84 ⋅ 0.4𝑛−1 for 𝑛 > 0) . (30) Channel 2 (initial 𝐼𝑆𝐼 = 0.88). The channel parameters were determined according to ℎ𝑛 = (0.4851, −0.72765, −0.4851) .

(31)

6

Mathematical Problems in Engineering 0

1

−2

0.8

−4

ISI (dB)

−6

0.6

−8 −10

𝐻 = 0.8

0.4

𝐻=1 𝐻 = 0.6

−12

0.2

−14

0

−16

−0.2

−18 −20

0

1000 2000 3000 4000 5000 6000 7000 8000 9000

−0.4

0

2

4

6

8

10

12

14

Iteration number Equalizer coefficients 𝐻 = 0.6

Simulated ISI Calculated ISI

Figure 2: A comparison between the simulated (with Godard’s algorithm) and calculated residual ISI for the 16QAM source input going through Channel 2 for SNR = 12 [dB]. The averaged results were obtained in 100 Monte Carlo trials. The equalizer’s tap length and step-size parameter were set to 13 and 0.00003, respectively.

Figure 4: Equalizer’s coefficients in the steady state obtained with Godard’s algorithm for the 16QAM source input going through Channel 2, SNR = 12 [dB] and step-size parameter equal to 0.00003. The equalizer’s tap length was set to 13.

0

0.8

−5

0.6 ISI (dB)

0.4 0.2

−10

𝐻=1

−15

𝐻 = 0.8

𝐻 = 0.6

0 −20

−0.2

−25

−0.4 −0.6

0

2

4

6

8

10

12

14

Equalizer coefficients 𝐻 = 1


The equalizer was initialized by setting the center tap equal to one and all others to zero. In the following we denote the residual ISI performance according to (6) with (8), (9), and (12) as “Calculated ISI.” Figures 2, 5, 8, 11, and 13 show the ISI performance as a function of the iteration number of our proposed expression (6) (with (8), (9), and (12)) compared with the simulated results for two different channels and SNR values, various values for 𝐻, and three different step-size values and equalizer’s length. According to Figures 2, 5, 8, and 13 (for 𝐻 = 0.6), a high

0

1000 2000 3000 4000 5000 6000 7000 8000 9000 Iteration number Simulated ISI Calculated ISI


correlation is observed between the simulated and calculated results. According to Figures 11 and 13 (for 𝐻 = 0.8), the Calculated ISI may be considered as an upper limit for the simulated results. Figures 3, 4, 6, 7, 9, 10, and 12 show the equalizer’s coefficients in the steady state for various values for 𝐻, SNR, equalizer’s tap-length, step-size parameters, and two different channels. According to simulation results (Figures 3, 4, 6, 7, 9, 10, and 12), it was reasonable to take approximately only √𝑁 − 1 instead of 𝑁 − 1 coefficients in (12) since indeed


7 0

0.8

−2 0.6

−4 −6 ISI (dB)

0.4 0.2

−10 −12

0

𝐻 = 0.6

𝐻 = 0.8 𝐻 = 0.9

−14

−0.2

−16

−0.4 −0.6

−8

−18 −20

0

2

4

6

8

10

12

14

0

2000

4000

6000

8000

10000

12000

Iteration number Equalizer coefficients 𝐻 = 1




1 0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

−0.2

−0.2

−0.4

0

2

4

6

8

10

12

14

Equalizer coefficients 𝐻 = 0.6


−0.4 −0.6

0

5

10

15

20

25

30



not all the coefficients in the equalizer’s tap length have significant values.

4. The Statistical Behaviour of the Residual ISI and Convolutional Noise The statistical behaviour of the residual ISI was never investigated in the literature. Furthermore, the convolutional noise 𝑝(𝑛) is usually assumed (in the latter stages of the deconvolution process [27]) to be a white Gaussian process.

In this section we investigate the statistical behaviour of the residual ISI and convolutional noise in the steady state. 4.1. The Statistical Behaviour of the Residual ISI. In this subsection we estimate the Hurst exponent of the residual ISI from the Rescaled Range (R/S) [28] with overlapping regions. For that purpose, let us denote 𝜉 as a vector of samples of the residual ISI (obtained from a single Monte Carlo trial) with

8

Mathematical Problems in Engineering 0.8

1

0.6 0.4

0.5

0.2 0 0

−0.2 −0.4 −0.6

0

5

10

15

20

25

30

−0.5

0


2

4

6

8

10

12

14




−2 −2

−4

−4

−6

−6

−8

−10 𝐻 = 0.6

−12

𝐻 = 0.8

𝐻=1

ISI (dB)

ISI (dB)

−8

−10

−14

−14

−16

−16

−18 −20

−18

0

𝐻 = 0.6

−12

0

2000

4000

1000 2000 3000 4000 5000 6000 7000 8000 9000 Iteration number


̃ The mean of 𝑛𝑝 consecutive samples in 𝜉 may be length 𝑁. defined as


̃ 𝑖𝑖 denotes the 𝑖𝑖th segment in vector 𝜉 and where 𝑛𝑝 + 𝑖𝑖 ≤ 𝑁, ⟨⋅⟩ means the estimate of (⋅). Next we define 𝑖+𝑖𝑖

𝛾𝑖,𝑛𝑝 ,𝑖𝑖 = ∑ [𝜉 [̃ 𝑢] − ⟨𝜉⟩𝑛𝑝 ,𝑖𝑖 ] , 𝑢̃=𝑖𝑖+1

𝑛𝑝 +𝑖𝑖

⟨𝜉⟩𝑛𝑝 ,𝑖𝑖

6000 8000 10000 12000 14000 16000 Iteration number



1 = ∑ 𝜉 [𝑖] , 𝑛𝑝 𝑖=𝑖𝑖+1

𝐻 = 0.8

𝑖𝑖 = 0, 1, 2, 3, . . . ,

(32)

𝑅𝑛𝑝 ,𝑖𝑖 = max (𝛾𝑖,𝑛𝑝 ,𝑖𝑖 ) − min (𝛾𝑖,𝑛𝑝 ,𝑖𝑖 ) ,

1 ≤ 𝑖 ≤ 𝑛𝑝 , (33) 1 ≤ 𝑖 ≤ 𝑛𝑝 .


9 1.04

1.06 1.04

1.02

1.02 1

1 0.98

0.98

0.96 0.96

0.94 0.92

0.94

0.9 0.88

0.92 0

20

40

Calculated 𝐻, 𝜇 = 0.00003 Average 𝐻, 𝜇 = 0.00003 Calculated 𝐻, 𝜇 = 0.00001

60

80

100

0

40

60

80

100

Average 𝐻, SNR = 20 dB Calculated 𝐻, SNR = 12 dB Average 𝐻, SNR = 12 dB

Calculated 𝐻, SNR = 30 dB Average 𝐻, SNR = 30 dB Calculated 𝐻, SNR = 20 dB

Average 𝐻, 𝜇 = 0.00001 Calculated 𝐻, 𝜇 = 0.00009 Average 𝐻, 𝜇 = 0.00009

Figure 14: Hurst exponent results (belonging to the ISI) for different step-sizes by using Godard’s algorithm, a 16QAM source input going through Channel 2, a 13-tap-length equalizer and SNR = 12 [dB] ̃ ≃ 9000 (the added noise was a white Gaussian process (𝐻 = 0.5)). 𝑁 and 𝑛𝑝 = [100 200 300 400].

20

Figure 16: Hurst exponent results (belonging to the ISI) for different SNR values (the added noise was a white Gaussian process (𝐻 = 0.5)) by using Godard’s algorithm, a 16QAM source input going through Channel 2, step-size parameter 𝜇 = 0.00001, and with a ̃ ≃ 9000 and 𝑛𝑝 = [100 200 300 400]. 13-tap-length equalizer. 𝑁

1.06 25 20 15 10 5 0 0.92

1.04 1.02 1

0.94

0.96

1

1.02

1.04

(a)

0.98 0.96 0.94 0.92

0.98

0

20

40

Calculated 𝐻, tap length = 19 Average 𝐻, tap length = 19 Calculated 𝐻, tap length = 27

60

80

100

Average 𝐻, tap length = 27 Calculated 𝐻, tap length = 13 Average 𝐻, tap length = 13

Figure 15: Hurst exponent results (belonging to the ISI) for different equalizer’s tap-length by using Godard’s algorithm, a 16QAM source input going through Channel 2, step-size parameter 𝜇 = 0.00001 and SNR = 12 [dB] (the added noise was a white Gaussian process ̃ ≃ 9000 and 𝑛𝑝 = [100 200 300 400]. (𝐻 = 0.5)). 𝑁

The Rescaled Range (𝑅/𝑆) [28] for the 𝑖𝑖th segment with 𝑛𝑝 samples in the vector 𝜉 is defined as [𝑅/𝑆]𝑛𝑝 ,𝑖𝑖 =

𝑅𝑛𝑝 ,𝑖𝑖 𝑆𝑛𝑝 ,𝑖𝑖

20

20

15

15

10

10

5

5

0 0.9

0.95

1

1.05

1.1

(b)

0 0.85

0.9

0.95

1

1.05

(c)

Figure 17: Histogram of the Hurst exponent (belonging to the ISI) for different step-size values by using Godard’s algorithm, a 16QAM source input going through Channel 2, and a 13-tap-length equalizer. ̃ ≃ 9000 and 𝑛𝑝 = [100 200 300 400]. The upper, bottom 𝑁 left, and bottom right figures were obtained for 𝜇 = 0.00001, 𝜇 = 0.00003, and 𝜇 = 0.00009, respectively.

where

𝑛𝑝 +𝑖𝑖

,

(34)

𝑆𝑛𝑝 ,𝑖𝑖

2 1 =√ ∑ (𝜉 [𝑖] − ⟨𝜉⟩𝑛𝑝 ,𝑖𝑖 ) . 𝑛𝑝 𝑖=𝑖𝑖+1

(35)

10


0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0

20

40

60

Calculated 𝐻, 𝜇 = 0.000005 Average 𝐻, 𝜇 = 0.000005 Calculated 𝐻, 𝜇 = 0.00001

80

100

Average 𝐻, 𝜇 = 0.00001 Calculated 𝐻, 𝜇 = 0.00009 Average 𝐻, 𝜇 = 0.00009

Figure 18: Hurst exponent results (belonging to the convolutional noise) for different step-sizes by using Godard’s algorithm, a 16QAM source input going through Channel 2, the noiseless case, and a 13̃ ≃ 9000 and 𝑛𝑝 = [100 200 300 400]. tap-length equalizer. 𝑁

The averaged Rescaled Range over multiple regions of the data with length 𝑛𝑝 is defined as [28] 𝐸[

𝑅𝑛𝑝 𝑆𝑛𝑝

]=

max(𝑖𝑖)

1 ∑ [[𝑅/𝑆]𝑛𝑝 ,𝑖𝑖 ] ≃ 𝐶𝑛𝑝𝐻, 1 + max (𝑖𝑖) 𝑖𝑖=0

(36)

where 𝐶 is a constant. Now, by applying the log2 operator on both sides of (36) we obtain [28] log2 (

max(𝑖𝑖)

1 ∑ [[𝑅/𝑆]𝑛𝑝 ,𝑖𝑖 ]) ≃ log2 𝐶 + 𝐻log2 𝑛𝑝 . 1 + max (𝑖𝑖) 𝑖𝑖=0 (37)

̃ This expression (37) may be seen as a line 𝑦̃ = 𝑏 + 𝑎𝑥, max(𝑖𝑖) where 𝑦̃ = log2 ((1/(1 + max(𝑖𝑖))) ∑𝑖𝑖=0 [[𝑅/𝑆]𝑛𝑝 ,𝑖𝑖 ]), 𝑥̃ = log2 𝑛𝑝 , 𝑏 = log2 𝐶 and the slope of the line is 𝐻 [28]. In order to estimate the Hurst exponent, several values for 𝑛𝑝 are needed. Figure 14 presents the Hurst exponent value estimated for different step-size values. For each step-size value we used 100 Monte Carlo trials where for each trial the Hurst exponent was estimated. Thus, we have for each step-size value 100 estimated values for the Hurst exponent. According to Figure 14, we observe two things. (1) The averaged Hurst exponent (obtained from 100 results) is higher (closer to one) for a lower step-size value. (2) The averaged Hurst exponent obtained for the different step-size values is very high (above 0.9). This means that the residual ISI is trending. If there is an increase from time index 𝑛 − 1 to time index 𝑛, there will probably be an increase from time index 𝑛 to time index 𝑛 + 1. The same is true of decreases, where a decrease will tend to follow a decrease. The large Hurst exponent value indicates that the trend is strong. In our case,

we see according to Figure 14 that the trend is stronger for lower step-size values. Figures 15 and 16 present the Hurst exponent value estimated for different equalizer’s tap lengths and SNR values, respectively. For each tap-length and SNR value we used 100 Monte Carlo trials where for each trial the Hurst exponent was estimated. Thus, we have for each tap-length and SNR value 100 estimated values for the Hurst exponent. According to Figures 15 and 16, the equalizer’s tap-length and SNR have nearly no impact on the Hurst exponent estimated from the residual ISI series. Figure 17 describes the simulated histogram of the estimated Hurst exponent for three different step-size parameters. According to Figure 17, the histogram of the estimated Hurst exponent resembles the Gaussian shape. 4.2. The Statistical Behaviour of the Convolutional Noise. In this subsection we estimate the Hurst exponent of the real part of the convolutional noise from the Rescaled Range (R/S) [28] with overlapping regions. For that purpose, let us denote now 𝜉 as a vector of samples of the real part of the convolutional noise (obtained from a single Monte Carlo trial in the convergence region for the noiseless case) with ̃ Please note that the real and imaginary parts of length 𝑁. the convolutional noise are independent. Thus, the statistical behavior of the real and imaginary parts of the convolutional noise is approximately the same. In the following we use (37) for estimating the Hurst exponent. Figure 18 presents the Hurst exponent value estimated for different step-size values. For each step-size value we used 100 Monte Carlo trials where for each trial the Hurst exponent was estimated. Thus, we have for each step-size value 100 estimated values for the Hurst exponent. According to Figure 18, we observe that as we enlarge the step-size parameter, the estimated Hurst exponent is closer to 0.5, while for lower values for the step-size parameter, the Hurst exponent is smaller than 0.5. According to [29], a Hurst Exponent value 𝐻 between 0 and 0.5 exists for time series with “antipersistent behaviour.” This means that an increase will tend to be followed by a decrease (or a decrease will be followed by an increase). This behaviour is sometimes called “mean reversion” which means future values will have a tendency to return to a longer term mean value. The strength of this mean reversion increases as 𝐻 approaches 0. As it was already mentioned earlier in this paper, the convolutional noise is often considered as a white Gaussian process. But a white Gaussian process has a Hurst exponent value of 0.5 which in our case is achieved only for high values for the step-size parameter.

5. Conclusion In this paper, we proposed a closed-form approximated expression (or an upper limit) for the residual ISI obtained by blind adaptive equalizers valid for the fGn input case where the Hurst exponent is in the region of 0.5 ≤ 𝐻 < 1. According to simulation results, a high correlation is obtained between the calculated and simulated results for the residual ISI for some cases while for others, the new obtained

Mathematical Problems in Engineering expression is a relative tight upper limit for the averaged residual ISI results. In this paper we investigated the statistical behavior of the residual ISI as well as the convolutional noise. We have found that the Hurst exponent of the residual ISI is close to one, almost independent of the SNR or equalizer’s tap length but depends on the step-size parameter. Since the Hurst exponent of the residual ISI is above 0.5, the residual ISI is trending. This means that if there is an increase from time index 𝑛 − 1 to time index 𝑛, there will probably be an increase from time index 𝑛 to time index 𝑛 + 1. The same is true of decreases, where a decrease will tend to follow a decrease. Concerning the convolutional noise, we have found that the convolutional noise obtained in the steady state is a Gaussian noise process having a Hurst exponent depending on the step-size parameter. Only for large values for the stepsize parameter we obtain approximately a white Gaussian process (𝐻 = 0.5) which is the statistical model assumed in the literature for the convolutional noise for the entire range of step-size values.

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